Question

let X be a binomial random variable with n=8 and p=0.6. find the following

(a) the mean of X

(b) the standard deviation of X

(c) P(X<4)

(d) P(X>5)

(e) P(3<X<6)

Answer #1

X ~ Binomial (n,p)

Where n = 8 , p = 0.6

Binomial probability distribution is

P(X) = ^{n}C_{x} p^{x}
(1-p)^{n-x}

a)

Mean = n * p

= 8 * 0.6

= **4.8**

b)

Standard deviation = Sqrt( np(1-p) )

= sqrt( 8 * 0.6 * 0.4)

= **1.38564**

c)

P( X < 4) = P( X <= 3)

= P( X = 0) +P( X = 1) +P( X = 2) +P( X = 3)

= ^{8}C_{0} 0.6^{0} 0.4^{8}
+^{8}C_{1} 0.6^{1} 0.4^{7}
+^{8}C_{2} 0.6^{2} 0.4^{6}
+^{8}C_{3} 0.6^{3} 0.4^{5}

= 0.1737

d)

P( X > 5) = P( X >= 6)

= P( X = 6) + P( X = 7) + P( X = 8)

= ^{8}C_{6} 0.6^{6} 0.4^{2}
+^{8}C_{7} 0.6^{7} 0.4
+^{8}C_{8} 0.6^{8} 0.4^{0}

= **0.3154**

e)

P( 3 < X < 6) = P( X = 4) + P( X = 5)

= ^{8}C_{4} 0.6^{4} 0.4^{4}
+^{8}C_{5} 0.6^{5} 0.4^{3}

= **0.5109**

Let X be a binomial random variable with n =
8, p = 0.4. Find the following values. (Round your answers
to three decimal places.)
(a)
P(X = 4)
(b)
P(X ≤ 1)
(c)
P(X > 1)

Let X represent a binomial random variable with n = 170 and p =
0.6. Find the following probabilities. (Do not round intermediate
calculations. Round your final answers to 4 decimal places.)
Probability
a.P(X ≤ 100)
b.P(X > 110)
c.P(105 ≤ X ≤ 115)
d.P(X = 90)

Let X be a binomial random variable with n = 10 and p = 0.2.
Find the following values. (Round your answers to three decimal
places.) (a) P(X = 4) (b) P(X ≥ 4) (c) P(X > 4) (d) P(X ≤ 4) (e)
μ = np μ = 2.00 (correct) (f) σ = npq σ = 1.265 (correct)

Let X be a binomial random variable with p = 0.7
a) For n = 3, find P(X=1)
b) For n = 5, find P(X≤3)

Let X represent a binomial random variable with
n = 180 and p = 0.25. Find the following
probabilities. (Do not round intermediate calculations. Round your
final answers to 4 decimal places.)
a. P(X ≤ 45)
b. P(X = 35)
c. P(X > 55)
d P(X ≥ 50)

Let X represent a binomial random variable with n = 180 and p =
0.25. Find the following probabilities. (Do not round intermediate
calculations. Round your final answers to 4 decimal places.)
a. P(X ≤ 45)
b. P(X = 35)
c. P(X > 55)
d P(X ≥ 50)

Let X represent a binomial random variable with
n = 360 and p = 0.82. Find the following
probabilities. (Do not round intermediate calculations.
Round your final answers to 4 decimal places.
Probability
a.
P(X ≤ 290)
b.
P(X > 300)
c.
P(295 ≤ X ≤ 305)
d.
P(X = 280)

1) Let X be a binomial random variable with p = 0.55, and n= 40.
Find the probability
that X is more than 2 standard deviations from the mean of X.

Please include steps.
Let x be a binomial random variable with n=4 and p=0.8. Find the
following. P(X = 3) P(X < 4)

if x is a binomial random variable, compute P(x) for each of the
following cases (record your answer to atleast 3 decimal
places):
(a) P(x≤1),n=3,p=0.5
P(x)=?
(b) P(x>4),n=8,p=0.6
P(x)= ?
(c) P(x<4),n=6,p=0.7
P(x)= ?
(d) P(x≥3),n=7,p=0.3
p(x)=?

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